problem archive
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Express the following in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). (i) 0.6 (ii) 0.47 (iii) 0.001 -
Express 0.99999... in the form \( \frac{p}{q} \). -
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of \( \frac{1}{17} \)? Perform the division to chec... -
Write three numbers whose decimal expansions are non-terminating non-recurring. -
Find three different irrational numbers between the rational numbers \( \frac{5}{7} \) and \( \frac{9}{11} \). -
Classify the following numbers as rational or irrational: (i) \( \sqrt{23} \) (ii) \( \sqrt{225} \) (iii) 0.3796 (iv) 7.478478... (v) 1.10100100010000... -
Check whether \( \frac{7}{\sqrt{5}}, \sqrt{2} + 21, \pi - 2 \) are irrational numbers or not. -
Add \( 2\sqrt{2} + 5\sqrt{3} \) and \( \sqrt{2} - 3\sqrt{3} \). -
Multiply \(6\sqrt{5}\) by \(2\sqrt{5}\). -
Divide \(8\sqrt{15}\) by \(2\sqrt{3}\). -
Simplify the following expressions: \( (5 + \sqrt{7})(2 + \sqrt{5}) \) -
Simplify the following expressions: \( (5 + \sqrt{5})(5 - \sqrt{5}) \) -
Simplify the following expressions: \( (\sqrt{3} + \sqrt{7})^2 \) -
Simplify the following expressions: \( (\sqrt{11} - \sqrt{7})(\sqrt{11} + \sqrt{7}) \) -
Rationalise the denominator of \( \frac{1}{\sqrt{2}} \) -
Rationalise the denominator of \( \frac{1}{2 + \sqrt{3}} \) -
Rationalise the denominator of \( \frac{5}{\sqrt{3} - \sqrt{5}} \) -
Rationalise the denominator of \( \frac{1}{7 + 3\sqrt{2}} \) -
Classify the following numbers as rational or irrational: (i) \( 2 - \sqrt{5} \) (ii) \( (3 + \sqrt{23}) - \sqrt{23} \) (iii) \( \frac{2\sqrt{7}}{\sqr... -
Simplify each of the following expressions: (i) (3 + \sqrt{3})(2 + \sqrt{2}) (ii) (3 + \sqrt{3})(3 - \sqrt{3}) (iii) (\sqrt{5} + \sqrt{2})^2 (iv) (\sq...